1. Technical Field
This invention relates to the field of finding integer solutions of equations whose graphs are conic sections.
2. Description of the Prior Art
Integer solutions are required to equations whose graphs are conic sections or parts thereof. Conic sections are parabolas, hyperbolae, circles or ellipses. Factorization is the most obviously useful such context and is chosen for illustration. Factorization is the task of, given an integer N, finding the (unique) primes whose product is N.
The task of factorization of a given integer is notoriously difficult, to the extent of rendering computationally infeasible the extraction of factors of numbers beyond a certain size. This infeasibility is what makes some cryptographic systems secure; for example, RSA cryptography.
There are many existing algorithms to perform the task of factorization, but each suffers from an increase in computation time as the input integer increases. This increase in computation time suffered by existing algorithms is inherent in tacit assumptions made about the model of computation in which these algorithms run. Crucially, it is assumed that instructions must be executed sequentially and this seems to be responsible for the algorithms' computational complexity.
An exception to this assumption of sequential execution is a factorization algorithm run in a quantum computing environment, where parallel execution of commands avoids the calculation time problem. However, technological limitations tightly constrain the input values able to be factorized.
It is an aim of the present invention to provide a fast method of finding integer solutions of conic equations. One embodiment of this provides a fast method of factorization. Just as with traditional algorithms, there is a limit to the size of numbers that can be factorized; however, in contrast with traditional algorithms, the proposed solution suffers no increase in calculation time as the input number approaches this limit.